When you mean “for example,” use e.g. It is an abbreviation for the Latin phrase exempli gratia. When you mean “that is,” use “i.e.” It is an abbreviation for the Latin phrase id est. Either can be used to clarify a preceding statement, the first by example, the second by restating the idea more clearly or expanding upon it. Because these uses are so similar, the two abbreviations are easily confused. If you just stick with good old English “for example” and “that is” you won’t give anyone a chance to sneer at you. If you insist on using the abbreviation, perhaps “example given” will remind you to use “e.g.,” while “in effect” suggests “I.E.”
Going off on a tangent, usually means someone continued on with an argument or point being made until they eventually get to the point to where the initial argument or point trying to be made is lost.